FORMULAS FOR COLUMNS WITI-1
SIDE LOADS AND ECCENTRICITY
November 1950
No. 121782
UNITED STATES PEPARTMENT OF AGRICULTURE
FOREST SERVICE
,1FORESTRODUCTS LABORATORY
LIS
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
FORMULAS FOR COLUMNS WITH SIDE
Laos
ANT ECCENTRICITY1
By
LYMAN W. WO(D, Engineer
Forest Products Laboratory,` Forest Service
U. S. Department of Agriculture
Introduction
The design of columns with side loads and eccentric end loads is an
old problem that has reieived attention from many authors. General textbooks
such as those by Church d or Maurer and Withey, th and the more specialized
work of Salmon on columns are among the many publications that have dealt
with the subject. The treatment of short ' columns, considered as prisms and
with stresses due to deflection neglected, is relatively simple, but long
columns with deflection stresses are more complex.
Critical loads on centrally loaded long columns that fail by bending
can be calculated satisfactorily with the Euler formula. Additional formulas of satisfactory accuracy have been developed for long columns with eccentric end load or with certain s p ecific combinations of side load with eccentric end load.
These latter formulas, however, require rather cumbersome
trial calculations involving the secant of an angle that can be determined
only indirectly; they have the further disadvantage that a separate formula
must be develo p ed for each condition of side loading.
Recognizing the difficulty in applying the secant formulas to problems
of safe loads on long columns with side loads and, eccentricity, Newlin"
sought a simplified method of design. He succeeded in develo p ing a general
formula in which eccentricity is simply represented and in which any condition
-This is an amplification and explanation of an article of the same title by
J. A. Newlin, formerly Chief, Division of Timber Mechanics, Forest Products
Laboratory, published in Building Standards Monthly, December 1940. The
formulas are also given in "National Design Specification for Stress-grade
Lumber and Its Fastenings," published by the National Lumber Manufacturers
Association. Acknowledgment is made to C. B. Norris of the Forest Products
Laboratory staff for review and confirmation of Newlin's analysis.
?Maintained at Madison, Wis., in cooperation with the University of Wisconsin.
1-Church, Irving P., Mechanics of Engineering, New York, 1914.
4
--Maurer, Edward R. and Withey, Morton 6., Strength of Materials, New York,
1940.
..zalmon, E. H., Columns, London, England, 1921.
Rept.
liri432
-1-
Agriculture-Madison
of side loading can be expressed in terms of the bending stress induced by
it. His formula, while somewhat more approximate than the secant formulas,
has as much accuracy as exists in the present state of knowledge of the
properties of wood or of the conditions under which wood columns are loaded.
The formula is not difficult to apply and is believed to be highly useful
in the design of wood columns.
New-lin t s original report and published article were rather brief, and
a number of steps in his derivation , of the long-column formula were omitted.
The derivation of a similar formula for short columns was also omitted. The
present report has been prepared to fill in the omitted material and to
illustrate the application of the formulas by means of suitable examples.
Notation
(All ,units except ratios are in pounds, inches, or combinations
thereof)
A
C
E
F
I
=
=
=
=
=
area of the cross section of a column.
unit failing stress in compression parallel to grain.
Youngs modulus of elasticity of the column material.
unit failing stress in flexure.
moment of inertia of the cross section of a column around the
neutral axis. As used here, the neutral axis is perpendicular
to the direction of eccentricity or side load. In a rectanglit
lar column, I =
bd3
IC = value of the slenderness ratio L dividing intermediate from long
a
columns :: L = length of a column or, more specifically, unsupported length.
L'
= slenderness. ratio of a column. As used here, d is measured in
the direction of eccentricity or side loads.
M = generally, the bending moment On a'colttafrOM'eCdentricity or,
Side load, but is used here also in the more restricted sense
of bending moment caused by the portion of side load that is
independent of axial load.
= generally, the flexural stress induced in the outer fiber of a
column from eccentricity onside load, but used, here also in
the more restricted sense of stress in the outer fiber induced
by the portion of side load that is independent of axial load.
P = axial load (end load) on a column, whether centrally or eccentrically applied.
= direct compressive stress induced by axial load (end load) on a
column.
S = section modulus of the cross section of a column related to flexure in the direction of eccentricity or side loads. Section
modulus of a rectangular column is bd2 •
i
Rept . R1782
-2-
7/2 E I
W = critical Euler load on a pin-ended column. W
L2
b = width of a rectangular column.
c = allowable unit working stress in compression parallel to grain
for a column of the length under consideration with centrally
applied axial load and no side load; this may be a short-column,
intermediate-column, or long-column stress.
d = depth of a rectangular column, measured in the direction of eccentricity or side load.
= additional deflection at midlength of a long column due to axial
load.
e-= eccentricity, the distance from the center of gravity of the column
section to the center of gravity of the applied load. Eccentricity in this analysis is considered as being parallel to one
of the sides of a rectangular column.
6 1 , 6 2 , 6 3 = increments of deflection at midlength of a column in
Salmon's analysis.
f = allowable unit working stress in flexure, as in a simple beam with
transverse loading only.
fo = flexural stress corresponding to the total deflection at midlength
of a long column when all eccentric and side loads are on.
fl, f2, f3 = increments of flexural stress corresponding to increments
of deflection E l , 6 2 , and 63.
fe = fictitious flexural stress due to eccentricity of axial load assumed
to replace the original deflection of the column in Salmon's
analysis.
m,. n = exponents in the general form of the equation for strength under
combined loading.
w = critical Euler unit stress on a pin-ended column. w =-A: .
yo = total deflection at midlength of a column in Salmon's analysis
when all loads are on.
z = ratio of flexural to direct compressive stress; that is, gip_
divided by FA; more specifically, a ratio of flexural stress
to direct compressive stress when both result from the same
loading, so that the ratio remains constant while the load varies.
Analysis of Combined Stresses
Textbooks on strength of materials show how the combined stress on a
prism or short column due to axial and flexural loads is represented by the
sum M/S +
In this expression, M is bending moment, which may be induced
by direct-acting side loads, by eccentricity of the axial load, or by both.
For purposes of design in some structural materials, the sum M/S + FAI is not
permitted to exceed a safe stress value, assumed to be the same in bending as
in compression.
Where strength in bending is unequal to that in compression, as is the
case with wood, there is a problem of determining what is the ultimate
strength value for various combinations of load. It is recognized that the
strength value under combined stress lies generally somewhere between the two
values for separated stresses and is influenced by the relative proportions
Rept. 81782
-3-
of each. The relationship which describes the condition for failure under
combined stress may be of the interaction type
(48)n ...,
(1)
where MVE and VA are stresses applied in flexure and compression, and F and
C are the corresponding ultimate strength values under separate loading.
and C. are known. values, either. M'S or P/A can be calculated by assumP/A becomes the strength under
ing a value for the other. Then the sum MIS the assumed combination of load.
•6
Newlinand_Trayer- - made tests of clear Sitka spruce under combined
loading which showed the relation of the strength to the relative proportions
of bending : and axial etress. The,curve of maximum stress from figure 6 of
their report,,
obtained from these test results, is shown as a solid line
in figure 1.
•
The most simple equation of type (1) is with exponents m and n each
equal tb:UnitythiS'relation gives combined strength values in clear Sitka
spruce `indicated by the lower broken -line curve in figure 1. A much closer
agreeMent • with the test results is obtained if m = 2, giving the upper brokenline curve of figure 1.
From consideration of figure 1, it is apparent that strength under
combined stress is estimated very closely from the relationship
(Mlf) 2
+
=
(2)
This relationship, however, is somewhat complex to handle as a formula,
especially when Ma is broken up into two or three components representing
side loads and eccentricity of axial load. Newlin t s formulaal for short
columns indicate'that he used the form
M/2 +
F
C
(3)
which is simpler of application. This relationship (3) has also had extensive
use by othe r designers dealing with the, problem of, combined loads. Figure 1
shows that it gives .rather conservative estimates of strengths but not far in
error, with the lower ratios of bending to total stress such as are most
likely to occur in short columns.
In the case of long columns, Newlin•simplified his safe-loading formula
by assuming that the ultimate strength under combined load is equal to the
bending strength. This is explained as follows. Long columns are those in
the range inwhich the Euler formula applies. The Euler formula is based on
ewlin, J. A. and Trayer, G. W., Stresses in Wood Members Subjected to Combined Column and Beam Action, Forest Products Laboratory Report No. 1311,.
1941.
Rept. 81782
elasticity rather than strength and can be applied only if TIA is less than
the proportional-limit stress in compression. This stress is usually assumed
to be two-thirds of the compressive strength of wood,1 or 2C/3 by the notation
of this report. The shortest column in the Euler class will reach its maximum load at this stress. The unit flexural strength, being a property of the
material, is unaffected by the length of the column. When stress due to an
axial load on a long column equals 2C/3, and if it is assumed that the maximum stress developed under combined load is equal to the flexural strength,
the available bending stress is F - 2C/3, and the ratio of.bending stress to
total stress becomes
F - 2c/3
F
This is a minimum value for that ratio in long columns. In the case of clear
Sitka spruce (fig. 1), the minimum ratio is
7,900 - 2,867
7,900
0.64
It follows that the long-column formula leads to ratios within the range of
0.64 to 1.00, as indicated by the heavy horizontal line at the top of figure 1.
Corresponding ratios based on safe rather than ultimate stresses and applicable
to other species range from about 0.55 to 1.00, as indicated on figure 1. It
can be seen that, in the range from 0.55 to 1.00, the maximum stress actually
developed in test is not much short of the flexural strength.
If the relation expressed in equation (2) were used to estimate maximum stress in a long column, a formula of considerable complexity would result. Equation (3) would give a usable formula, but would result in overly
conservative estimates of load. The assumption that strength under combined
loading of long columns equals flexural strength has therefore been chosen.
Short Columns
Wood columns with slenderness ratios (ratio of unsupported length to
least dimension of cross section) of 11 or less receive both compression and
flexural stresses under eccentric axial load, but it is not necessary to
assume any additional stress due to deflection. The combined stress, derived
4 is expressed by the
as for prisms by well-known principles of mechanics, –
quantity EL 4. mil . In this expression M is bending moment, which may be
induced by eccentricity of an axial load, by direct-acting side loads, or by
both. For maximum safe load, the relationship of equation (3) is used with
safe stresses instead of ultimate strength values.
=1
(4)
C
Equation (4) is directly applicable where there are side loads and
concentric axial load. If the end load is proportional to the side load (as
'Newlin, J. A. and Gahagan, J. M., Tests of Large Timber Columns and Presentation of the Forest Products Laboratory Column Formula. U. S. Dept, of
Agr. Tech. Bull. No. 167, 1930.
Rept. 81782
-5-
in upper chord members of roof trusses, with bending stress induced by roof
loads acting through purlins, and with axial loads proportional to the same
roof loads), 112S can be replaced by z P/A, and equation (4) becomes
.
z (P/A)
_
f
(5)
c
The bending 'stress MIS may be induced by eccentricity of the axial
load instead of by side load. In that case, the external moment M is expressed by the quantity, Pe. The section modulus S may be replaced by
71.7f (d measured in the direction of eccentricity). Then
„M/SI, Ped P (Aed‘
= ‘TT
(6)
bd3
But in a rectangular column, A = bd and I = ---. Substitution of these values
12
in (6) gives
6e
M/S = P/A (--)
(7)
Substitution of expression (7) in equation (4) gives
P/A (6 ejd)
=
(8)
c
f
for the condition of eccentric axial load and no side load.
Where both side load and eccentric axial .load are present the term
M/S
In
(4) represents the total of bending stress from both. Furthermore,
-—
the side load may be considered as composed of two portions, one independent
of axial load
and the other proportional to axial load (z ILA). The
total bending stress is thus expressed by
(E/1)
P/A
6e
d-
+ M/S + z P/A
and equation (4) becomes
P/A
(6 e/d) + M/S ,+ z P/A
C.
=
(9)
where yls has the more restricted meaning of bending stress due to that portion of side load which is independent of axial load.
Equation (9) is a general equation from which any of the others is
readily derived by dropping certain terms. For example, if all side load is'
proportional to axial load, and axial load is eccentrically applied, the term
M/S is dropped. If all side load is proportional to axial load, and axial
load is centrally applied, both sand ILA (-6e
-) are dropped, and equation (5)
results. If there is eccentricity but no side load, MIS and z P/A are dropped,
Rept. R1782
-6-
and equation (8) results. Other loading conditions can be similarly represented by suitable selection of terms. Where there is neither side load
nor eccentricity, all three terms in the numerator of the first fraction of
(9)
disappear, leaving only
= 1 or
12/1s
= c.
,
6e
It is to be noted that the term P/A
(--) in equation
•
•
a
(9)
is developed
from consideration of a rectangular cross section. Equation (9) is therefore
applicable in this form only to columns of square or rectangular cross
section.
Any of the foregoing equations can be solved, either for PIA or for
M/S, if desired to facilitate any particular design problem. Solution for
PTA or. MIS will be easier if any redundant terms are first dropped. Newlin
published these equations in a form in which they are solved for f/21,1
Long Columns
In long columns of the Euler class, the problem of eccentric and side
loads is made more complex by the addition of stress induced by the curvature of the column itself. The secant formulas, are applicable but rather
difficult to use in problems of determination of safe load. Furthermore, it
is desirable to express all conditions of side loading in terms of their
resulting moments or stresses, thus making one general formula applicable to
all loading conditions. Newlinl succeeded in doing this.
In developing a general formula fcr the condition of side and eccentric axial loads, use is made of a relationship developed by Salmon for
columns bent to sinusoidal curvature. While it is recognized that the
most common side loadings do mot produce sinusoidal curvature, Salmon showed
that with small amounts of curvature, the deflection is practically the same
whether the elastic curve is circular, parabolic, or sinusoidal. The error
introduced by assuming sinusoidal curvature is probably of lesser magnitude
than approximations . in the present state of knowledge on the strength properties of wood or on the conditions under which wood columns are loaded.
Salmon considered a column originally bent to a sinusoidal curvature
with a deflection el at midlength (fig. 2). Under the action of an axial
load P, the deflection increases to a value yo. The original curvature of
the column is represented by the equation —
y = e l cos x
For this condition, Salmon arrived at the relationship
61
Yo - 1
Rept. 81782
-7-
p/w
(10)
where P is the axial load on the column and W is the critical Euler load
(notation not the same as that.used by.Salmon). By algebraic transformation,
equation (11) becomes
el P =
1
(12)
One property of a sinusoidal elastic curve is that deflection is proportional to the stress Causing it. sking, for example, the equation for
a sinusoidal curve (10), there follows by differentiation
= _
dx2
a
oos2x * -
L2
L
2
L2
y
(13)
Since this is an elastic curve, the general equation of external and internal
moments in bending may be applied:
(14)
2
Substituting ..–Zfrom equation (13)
dx2
= EI
,,,2
L2
(15)
y
From the general expression for bending stress equal to
stress
=
vt
"2
S L2
y
LS,
(16)
Equation (15) shows the proportionality of moment, and equation (16) shows
the proportionality of stress to deflection.
Newlin used Salmon's originally bent column (fig. 2) to represent an
eccentrically loaded long column with axial load P and an eccentricity corresponding to the deflection El. If side loads are applied to such a column,
additional deflections e2 and T3 may result. If curvature remains sinusoidal,
the summation of deflections E l f 2 51 will hold the same relationship to
yo as El holds in equation (1TT. Then
•••nn••••
1 4.
+ !3
yo
11'4
7
(1 )
It has already been shown in equation (16) that, with sinusoidal
curvature, deflection is proportional to . stress. Since each of the curves in
figure 2 is assumed to be sinusoidal, all of the deflections e ll !2 , and !„.1,
and yo have the same factor of proportionality to the bending stresses that
Rept, R1782
-8-
cause them. The deflection ratio
62
61
+ 63
yo
is thus equal to a stress ratio
f l f2
f3
fo
The load ratio p/W can also be replaced by the corresponding stress
ratio
where w is the critical Euler stress. Equation (17) then becomes
fl
f2
f3
=1
(18)
fo
Now let fl be re placed by a fictitious stress fe , which, if it had
been present in Salmon's column, would have caused the deflection e l . The
fictitious stress f e which could have caused deflection e l can be evaluated
in terms of the critical Euler load by substituting fe and el in equation
(16) for a sinusoidal elastic curve
EI
(19)
It is now necessary to determine what eccentricity e on a straight
column corresponds to the deflection e i in Salmon's originally bent column.
The two conditions are compared in figure 3. Consider Salmon's column (fig.
3A) with an original deflection e l and bent under a small load P to an
additional deflection S with sinusoidal curvature. From equation (12)
1
(2 0)
By algebraic transformation
F
p/w
8 =E
ll 1 - P/WJ
(21)
which, when LW' is small, becomes approximately
8
= E
(P /W)
(22)
If the column were originally straight with the same load P applied
with an eccentricity e such that the same deflection-Vresulted (fig, 3BY,
there is obtained from the secant formula for eccentric load
Rept. E1782
-9-
L P
e(sec \it;
- 11= e(sec CI
PE I
Since in the Euler formula
r2 El
W -4 =
Substituting
(24) in
L2
and
1)
(23)
IT2
MY UT
(24)
(23) gives
= (sec
\FR -
1)
( 25 )
Tables of integrals-8 express the secant of an angle in terms of the series
sec x = 1
4-
x2 + %,21 + 61x6
6:
( 26)
With LW small, terms beyond the first two of (26) are negligible, and (25)
becomes very nearly
$
e (1+
/72p
- 1) =e (kr-)
(27)
Equating (27) with (22) gives
e
is very nearly,
Since
ri
r2p
e t1
(28)
(29)
=
Newlin estimated the relationship of eto_ci from a different basis11
but arrived at the same values as in (29), pointing out that e l could be
rep1ated by 2 with results that imuld;check_quite closely with the secant
f
formula.
Substitution of
(29) in (19) gives
Er tr2
e
(30)
4sL2
Substitution of expression (24) for the critical Euler load on a pinended column in (30) gives
(51)
fe = wA ( e)
ierce,
1929.
Rept. 81782
A Short Table of Integrals, third edition revised, New York,
be
In a rectangular column, A = bd and S =–6– (d measured in the direction of eccentricity of the load), from which
fe=w(15g)
(32)
Now let f2 in equation (18) be replaced by MZ2, a stress induced by a side
load that is independent of the axial load: Let f be replaced by z P/A, a
stress induced by a side load that is proportional to the axial load. Note
that either AIL or z P/A may be entirely missing in actual loading conditions.
Substituting the equivalents for f l =fo , f, and !: in equation (18)
gives
w 456:
) + M/S + z
` 2d
•
fo
=1
(33)
Equation (33) expresses the condition for breaking load, since w is
the Euler critical or breaking stress. The same relation holds for safe or
design load by substituting c, the Euler safe working stress, for w. Equation
(33) then becomes
c (122 ) + M/S + z P/A p/A
+
fo
=1
(34)
The available value for the bending stress f o when all loads are on is
now limited by the safe bending stress value f. In the extreme fiber on the
concave side of:the column, f o and P/A are additive, and their sum cannot be
permitted to exceed f (as previously noted under "Analysis of Combined
Stresses"). In the analysis leading to equation (11), f„, was not applied, so
for Salmon's relationship to hold, e 'or its equivalent c (
) should be de2d
ducted from the sum f o + LA. Then, for the condition of safe loading, the
13e
sum f + P/A c () cannot exceed f. To solve for the maximum safe load
– 2d
using the assumption developed under "Analysis of Combined Stresses,"
f o + P/A c (122 ) =f
2d
By transposition, fo = f - P /A' + c 122 and equation (34) then becomes
2d
c 13i)
+ M/S + z P/A P/A
f - P/A + c(lii)
Rept. E1782
-11-
(35)
Algebraic transformation of equation (35) by clearing fractions and
collecting terms gives a more simple form
P/A (lig)
f
z P/A
PA
_A =1
( 36 )
the general equation for the maximum Safe value of any combination of side
loads and eccentricity on long columns.
Equation (36) is in a general form from which equations for a number
of special cases can be readily derived by dropping the inapplicable terms.
For example, if all side-woad is proportional to axial load, and axial load
is eccentrically applied, the term lig is dropped. If all side load is proportional to axial load, and axial load is centrally applied, both Ea and
( 22 ) are dropped. Where there is eccentricity but no side load, both
ma
and z SP/A) are dropped. -Other loading conditions can be similarly represented by suitable selection-of terms. Where there is neither side load nor
eccentricity, all three terms in the numerator of the first fraction of (36)
These are the same processes as
a =
= 1 or,.74
disappear, leaving only in equation (9) for short columme; it is to be remembered, however, that c
in equation (36) is the safe long-column stress, while c in equation (9) is
the safe short-column stress.
Because-f the method of its derivation equation (36), like equation
(9) 1 40 applicable only to columns of square or rectangular cross section.
Equation (36) or any of the specialized equations derived from it can
be solved for ELA or for N115, if desired to facilitate any particular design
problem. It is to-.be noted that solution for TA results in most cases in a
quadratic equation, for which there are two roots of the form
* Vb2 . 4ac
2a
The root using the minus sign before the radical is used, since the root
using the plus sign would give an impossible result, Newlin published these
equations in a form in which they are solved for P/A.1
Equation (36) is developed from the assumption that maximum moment and
Maximum geflection occur at midlength of the column. This is true for eccentric axial load and for side load applied symmetrically along the length, but
is not true with large aide loads unsymmetrically placed along the length of
the column. Recognizing this point, Newlin made the following recommendations
in regard to unsymmetrically applied side load.
1. For a single concentrated side load, the stress under the load
can be used, regardless of the position of the side load with reference to
the length of the column. Flexural stress from side load is maximum at this
point.
Rcpt. R1782
-12-
2. T11# stress to use with a system of side loads is the maximum
stress due to the system. Where the system of side loads is such that maxi
mum moment from side load is present near the end of the column, some slight
error on the side of overload will occur.
Columns of Intermediate Length
Newlin chose a lower limiting value of 20 for the slenderness ratio.
() in his long-column formula (36). This limit was chosen as being approxid
mately the lower limit for values of K (slenderness ratio separating
intermediate- and long-column groups) ^in the most common species and grades
for structural use. In many species and grades, the K value exceeds 20, and
formula (36) may be used for intermediate columns coming under the Forest
Products Laboratory fourth-power parabolic formula' instead of the Euler
formula. Where this is the case, the value of c in equation (36) is determined from the parabolic instead of the Euler formula, but use of equation
(36) is otherwise the same.
,L
For columns with slenderness k a) ratios between 11 and 20, the safe
loadings under side load and eccentricity may be determined by straight-line
interpolation between equation (9) for
-a:
11 and equation (36) for a- = 20.
Loadings thus determined may be somewhat in error on the conservative side.
Examples
Assume a species and grade of wood having the following properties:
E . 1,600,000 pounds per square inch
f = 1,600 pounds per square inch
c = 1,200 pounds per square inch (for short column)
Values of c for intermediate or long columns are obtained with the
Wood Handbook formulas ..9
Assume a 6- by 8-inch (nominal dimensions) column in various lengths
and under various conditions of loading. The side loads and eccentricity are
assumed as acting to bend the column in the 8-inch direction, The actual
cross-sectional dimensions are 5-1/2 by 7-1/2 inches.
The design of the column in these examples provides only for flexure
in the 8-inch direction. The column is assumed to be stable or supported so
that there is no deflection in the 6-inch direction; if simultaneous deflection
2Forest Products Laboratory Wood Handbook. U. S. Dept. of Agr., revised 1940,
p.163.
Rept. R1782
-13-
in both directions is possible, the analysis becomes , more complex. For the
purpose of determining limits of application of, the formulas in the examples,
the slenderness ratio isbased on 7-1/2 rather than 5-1/2 inches.
Long Column with Side Load and Concentric Axial Load
Let the column be 20 feet long, so that = 32. Assume a side load of
75 pounds per foot of length, uniformly distributed. Determine the safe axial
load centrally applied.
Equation (36) is utilized by dropping the terms representing eccentricity and side load proportional to end load, So that it becomes
M/S
f - R JA
±
P/A
L.-- 1
c
Since WS is known, the equation is solved for ILA giving
Pik ( f 2
-
V( ±:—.4r- C ) 2 7 c(f m/0
(37)
_ 75. x 20„x_20 x 12 x 6 x 4 x 2 _873 1).s.i.
8 x 11 x l5 x 15
M/S
c -
0.274 x 1,60q1000
32 x 32
f + c 1,600 428
2 -
2
428 pis.i.
1,014
= 1,014 - q ( 1, 014 ) 2 - 428 (1,600 - 873) = 167 P.s.1.
The safe load is 5.5 x 7.5 x 167 - 6,900 pounds.
Eccentric Axial Load on Long Column
Let the column be'20 feet long, and the eccentricity be 2-1/2 inches,
with no side load.
Equation (36) takes the form
p
( .4221
1
2d/ 4.
/
f P/A
Rept. R1782
-14-
which, solved for
P/A
ILA gives
f + c (1 +.15-2)
2d
2
f+ c (1 +) 2
-
2
L
cf
(38)
f = 1,600 p.s.i. and c = 428 p.s.i. as before
(1
2d'
I
P/A =
= j. 15 x 5 x 2
' 2 x 15 x 2 3.5
+3.5c -
1,549
2
1,549 - V(1,549) 2 - (428)
The safe load
is
5.5 x 7.5
(1,600) = 240 p.s.i.
x 240 = 9,900 pounds.
If the length of the column is 12-1/2 feet, so that = 20, the solud
tion is the same except for the value of c. Since K = 23.4, which is greater
than 20, the fourth-power parabolic formula is used, and c = 986 pounds per
square inch. Substitution of this value in equation (38) gives P/A = 335
pounds per square inch. The safe load is 5.5 x 7.5 x 335 13,800 pounds.
Eccentric Axial Load on Short Column
Let the column be 7 feet long, so that
11. The eccentricity is
Q.
2-1/2 inches, and there is no side load. Determine the safe load.
Equation (8)
is P/A 6e
(--) P/A
d + — = 1.
When solved for P/A, this becomes
P/A -
cf
6e
f + c a-
5x2 1
c= 1,200 p.s.i., f = 1,600 p.s.i., e
d 15 x 2PIA 1,200 x 1,600
1,600 + 2,400
480 pounds per square inch.
The safe load is 5.5 x 7.5 x 480 19,800 pounds.
Rept. R1782
-15-
(39)
Eccentric Axial Load on Intermediate Column
If the column is 10 feet long, so that.( = 16, the safe load is ob= 11 and = 20.
tained by straight-line interpolation between the values for
a
19,800 - 5/9 (19,800 - 13,800) = 16,500 pounds.
Axial Load Known
Where P/A has a known value and it is desired to determine the permissible side load, equations (9) or (36) or any modifications thereof can be
solved for M/S. Since M/S appears only in the first power in either equation,
solution for it offers no special problems. Having determined ML, the permissible side load for the assumed conditions can be found.
Eccentricity and Two Side Loads
A 4- by 16-inch (nominal) member is used with width vertical in the
upper chord of a roof truss. It supports a number of uniformly distributed
purlin loads, with purlins assumed to give full, lateral support so that the
member qualifies as a short column. The roof system is designed so that the
flexural stress in the member from the roof loads through the purlins is onehalf the axial stress in the member from the truss reactions. There is in
addition a concentrated load suspended near the center of length of the member, causing a flexural stress of 200 pounds per square inch. The design at
panel points of the truss is such that the axial load is centered at a point
1.55 inches above the center of width of the member. Physical properties of
the material are the same as in the preceding examples. Determine the safe
axial load, assuming actual dimensions of 3-5/8 by 15-1/2 inches.
In this case, equation (9) in its complete form is used. When solved
for
LA, this becomes
(f M/S)
p/A f + c
PIA _
loads.
e
1.,5
d
15.5
_
(40)
+ z)
1
z-1
10
2
1,200 (1,600 - 200)
1,200 x l2400
- 525 p.s.i.
+
1)
1,600
+ 1,320
1,600 + 1,200
10
2
z P/A 525 - 262 pounds per square inch flexural stress from purlin
Safe axial load is 525 x 3-5/8 x 15-1/2 = 29,500 pounds.
Rept. R1782
-16-
Column Formula Summarized
The general formulas developed by Newlin are
P/A ( -§i) + M/S + z P/A P/A
(9)
for columns with slenderness ratios of 11 or less, and
P/A ( 122 ) + M/S + z P/A 2/1 ,
2d
f - F/A
=
(36)
for columns with slenderness ratios of 20 or more, where
P/A = direct compressive stress induced by axial load.
M/S = flexural stress induced by side loads that are independent of
end load.
c = allowable unit working stress in compression parallel to grain
for a column of the slenderness ratio under consideration
with centrally applied axial load and no side load.
f = allowable unit working stress in flexure that is permitted
where flexural stress only exists.
e = eccentricity.
d = depth of column, measured in the direction of side loads or
eccentricity.
z = ratio of flexural to direct compressive stress when both result
from the same loading, so that the ratio remains constant
while the load varies.
Stresses for columns with slenderness ratios between 11 and 20 are
determined by straight-line interpolation between formula (9) for a slenderness ratio of 11 and formula (36) for a slenderness ratio of 20.
These formulas may be simplified for some conditions of loading by
dropping out certain terms; for example, if there are side loads and a concentrically applied end load, e becomes zero, and the first term in the
numerator of equations (9) and (36) disappears. The formulas can be solved
for ELI!! or for M/S where this will facilitate their use.
These formulas are applicable only to columns of square or rectangular
cross section.
Where side loads are such that maximum deflection and maximum flexural
stress do not occur at midlength of the column, it is generally satisfactory
to consider MJ as the maximum flexural stress due to the load or loads,
regardless of its position in the length of the column. When the point of
this maximum stress is near the end of the column, a slight error on the side
of overload will occur.
Rept. R1782
-17-
Figure 1.--Strength values-. of clear Sitka spruce
under comblned,strasa; as: determined:bY test and
calculated by two methods.
Z.M 85826 F
.
Rept. 81782
.7
Figure 2.--Load on column with sinusoidal
curvature.
zm 85824 F
Rept. 81782
Figure 3.--Straight column with eccentricity
equivalent to Salmon's originally bent column,
A, Salmon's originally bent column; B, straight
column with eccentric loading.
ZM 85825 F
Rept. R1782